Lecture Notes: Ratio, Proportion, and Indices

Introduction

Welcome and Greetings
- Good evening to everyone!
- Introduction to the chapter on Ratio, Proportion, and Indices.
- Importance of understanding small concepts for solving questions.

Completion of previous chapters:
- Chapter 7: One Shot revision completed.
- Chapter 2: Equations One Shot completed.
Today's focus is on Chapter 1: Ratio, Proportion, and Indices.
- Four key topics will be covered today.
- Past year questions will also be practiced.

Definition:
- A ratio is a comparison of two quantities of the same kind and same unit.
- Example: Height comparison should be in the same unit (cm, meters, etc.).
Conditions for forming a ratio:
- Quantities must be of the same kind and measured in the same unit.
Types of Ratios:
- Terms:
  - a (first term or antecedent)
  - b (second term or consequent)
- Ratios should always be simplified (e.g., 12:1 should be written as 3:4).

Definition:
- Proportion states that two ratios are equal.
- Denoted as a:b :: c:d.
Cross Product Rule:
- If a:b :: c:d, then a*d = b*c.
Types of Proportions:
- Continued Proportion:
  - For three quantities a, b, c, if a/b = b/c, then they are in continued proportion.
- Mean Proportional:
  - The geometric mean of two numbers can be obtained through continued proportion.

Definition:
- Indices (or exponents) represent powers of numbers.
- Example: a^n means a is multiplied by itself n times.
Laws of Indices:
- If bases are the same, powers are added when multiplied:
  - a^m * a^n = a^(m+n)
- If bases are the same, powers are subtracted when divided:
  - a^m / a^n = a^(m-n)
- Power of a power:
  - `(a^m)^n = a^(m*n)
- Negative powers:
  - a^-n = 1/a^n.
- Fractional powers:
  - a^(1/n) represents the n-th root of a.

Practice Questions:
- Encourage students to solve practice questions from study material and past year papers.
Next Session:
- Focus on logarithms and a review of ratios, proportions, and indices.
Closing Remarks:
- Thank you for your participation and engagement!